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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(190))
M = H._module
chi = DirichletCharacter(H, M([57,70]))
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gp:[g,chi] = znchar(Mod(39, 9025))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.39");
| Modulus: | \(9025\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(9025\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(190\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{9025}(39,\cdot)\)
\(\chi_{9025}(134,\cdot)\)
\(\chi_{9025}(229,\cdot)\)
\(\chi_{9025}(419,\cdot)\)
\(\chi_{9025}(514,\cdot)\)
\(\chi_{9025}(609,\cdot)\)
\(\chi_{9025}(704,\cdot)\)
\(\chi_{9025}(894,\cdot)\)
\(\chi_{9025}(989,\cdot)\)
\(\chi_{9025}(1179,\cdot)\)
\(\chi_{9025}(1369,\cdot)\)
\(\chi_{9025}(1464,\cdot)\)
\(\chi_{9025}(1559,\cdot)\)
\(\chi_{9025}(1654,\cdot)\)
\(\chi_{9025}(1844,\cdot)\)
\(\chi_{9025}(1939,\cdot)\)
\(\chi_{9025}(2034,\cdot)\)
\(\chi_{9025}(2129,\cdot)\)
\(\chi_{9025}(2319,\cdot)\)
\(\chi_{9025}(2414,\cdot)\)
\(\chi_{9025}(2509,\cdot)\)
\(\chi_{9025}(2604,\cdot)\)
\(\chi_{9025}(2794,\cdot)\)
\(\chi_{9025}(2984,\cdot)\)
\(\chi_{9025}(3079,\cdot)\)
\(\chi_{9025}(3269,\cdot)\)
\(\chi_{9025}(3364,\cdot)\)
\(\chi_{9025}(3459,\cdot)\)
\(\chi_{9025}(3554,\cdot)\)
\(\chi_{9025}(3744,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
| Field of values: |
$\Q(\zeta_{95})$ |
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sage:CyclotomicField(chi.multiplicative_order())
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gp:nfinit(polcyclo(charorder(g,chi)))
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magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 190 polynomial (not computed) |
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sage:chi.fixed_field()
|
\((5777,3251)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{7}{19}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(39, a) \) |
\(1\) | \(1\) | \(e\left(\frac{127}{190}\right)\) | \(e\left(\frac{59}{190}\right)\) | \(e\left(\frac{32}{95}\right)\) | \(e\left(\frac{93}{95}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{1}{190}\right)\) | \(e\left(\frac{59}{95}\right)\) | \(e\left(\frac{36}{95}\right)\) | \(e\left(\frac{123}{190}\right)\) | \(e\left(\frac{173}{190}\right)\) |
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sage:chi(x) # x integer
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gp:chareval(g,chi,x) \\ x integer, value in Q/Z
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magma:chi(x)
